Is it true that for any non-orientable Riemannian surface of genus 2 there exists Conformal mapping of degree two to a projective plane?
Also, is the following argument works? Given any Riemann surface $M$ of genus two with a free involition $\sigma$ we can find a meromorphic $f$ function with the only pole of second order (existence of Weierstrass point). Then we consider the function $f +\sigma^*f$, it is a conformal mapping to a Riemann sphere of degree 4 and it descends to a factor $M/\sigma$. As a result we have a conformal mapping of order 2 of any non-orientable to a Riemann sphere. I have some suspicions about the result, but do not seem to find an error.